# Find the value of c and state the amplitude and period of f (?)

## the function f is defined, for all values of x, by $f \left(x\right) = 2 \sin \left(\frac{x}{3}\right) + c$ , where c is a constant. the graph $y = f \left(x\right)$ passes through the point $\left(\frac{\pi}{2} , - 3\right)$ (a) find the value of c (b) state the amplitude and period of f (c) sketch the graph of $f \left(x\right)$ for $0 \le x \le 3 \pi$

Aug 11, 2018

#### Answer:

Please see the explanation below

#### Explanation:

The function is

$f \left(x\right) = 2 \sin \left(\frac{x}{3}\right) + c$

The graph passes through the point $= \left(\frac{\pi}{2} , - 3\right)$

$- 3 = 2 \cdot \sin \left(\frac{\pi}{3} \cdot \frac{1}{2}\right) + c$

$- 3 = 2 \sin \left(\frac{\pi}{6}\right) + c$

$\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$

$c + 1 = - 3$

$c = - 4$

The value of $c = - 4$

The function is

$f \left(x\right) = 2 \sin \left(\frac{x}{3}\right) - 4$

The amplitude is $= 2$

The perod is $T = 2 \frac{\pi}{\frac{1}{3}} = 6 \pi$

See the graph below

graph{2sin(x/3)-4 [-2.54, 48.76, -12.15, 13.5]}